Function Restrictions and Extensions
Why
The relationship between the inclusion map and the identity map is characteristic of making small functions out of large ones.1
Definition
Let $X \subset Y$ and $f: Y \to Z$. There is a natural function $g: X \to Z$, namely the one defined by $g(x) = f(x)$ for all $x \in X$. We call $g$ the restriction of $f$ to $X$. We call $f$ an extension of $g$ to $Y$. Clearly, there may be more than one extension of a function
Notation
We denote the restriction of $f: Y \to Z$ to the set $X \subset Y$ by $f\mid X$ or $f_{\mid X}$.
Example
A simple example is the that the inclusion mapping from $X$ to $Y$ with $X \subset Y$ is a restriction of the identity map on $X$
An extension order
Here is a natural order involving set extensions and restrictions. Fix two sets $A$ and $B$. Let $F$ be the set of all functions $f: X \to Y$ with $X \subset A$ and $Y \subset B$. Define a relation $R$ in $F$ by $(f, g) \in R$ if $\dom f \subset \dom g$ and $f(x) = g(x)$ for all $x$ in $\dom f$. In other words, $(f, g) \in R$ if $f$ is a restriction of $g$ (or, equivalently, $g$ is an extension of $f$. We recognize that $R$ is a special case of the inclusion partial order by recognizing the elements of $F$ as subsets $A \times B$.
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